Time in model theory
The time order between interpretations of expressions
Given a model, expressions do not receive their interpretations all
at once, but only the ones after the others, because these
interpretations depend on each other, thus must be calculated after
each other. This time order of interpretation between expressions,
follows the hierarchical order from sub-expressions to expressions
containing
them.
Take for example, the formula xy+x=3. In order for it
to make sense, the variables x and y must take a
value first. Then, xy takes a value, obtained by multiplying
the values of x and y. Then, xy+x
takes a value based on the previous ones. Then, the whole formula
(xy+x=3) takes a Boolean value (true or false).
But this value depends on those of the free variables x and
y. Finally, taking for example the ground formula ∀x,
∃y, xy+x=3, its Boolean value (which is false
in the world of real numbers), «is calculated from» those taken by
the previous formula for all possible values of x and y,
and therefore comes after them.
A finite list of formulas in a theory may be interpreted by a single
big formula containing them all. This only requires to successively
integrate (or describe) all individual formulas from the list in the
big one, with no need to represent formulas as objects (values of a
variable). This big formula comes (is interpreted) after them all,
but still belongs to the same theory. But for only one formula to
describe the interpretation of an infinity of formulas (such as all
possible formulas, handled as values of a variable), would require
to switch to the framework of one-model theory.
The metaphor of the usual time
I can speak of «what I told about at that time»: it has a sense if
that past saying had one, as I got that meaning and I remember it.
But mentioning «what I mean», would not itself inform on what it is,
as it might be anything, and becomes absurd in a phrase that
modifies or contradicts this meaning («the opposite of what I'm
saying»). Mentioning «what I will mention tomorrow», even if I knew
what I will say, would not suffice to already provide its meaning
either: in case I will mention «what I told about yesterday» (thus
now) it would make a vicious circle; but even if the form of my
future saying
ensured that its meaning will exist tomorrow, this would still not
provide it today. I might try to speculate on it, but the actual
meaning of future statements will only arise once actually expressed
in context. By lack of interest to describe phrases without their
meaning, we should rather restrict our study to past expressions,
while just "living" the present ones and ignoring future ones.
So, my current universe of the past that I can describe today,
includes the one of yesterday, but also my yesterday's comments
about it and their meaning. I can thus describe today things outside
the universe I could describe yesterday. Meanwhile I neither learned
to speak Martian nor acquired a new transcendental intelligence, but
the same language applies to a broader universe with new objects. As
these new objects are of the same kinds as the old ones, my universe
of today may look similar to that of yesterday; but from one
universe to another, the same expressions can take different
meanings.
Like historians, mathematical theories can only «at every given
time» describe a universe of past mathematical objects, while this
interpretation itself «happens» in a mathematical present outside
this universe.
Even if describing «the universe of all mathematical objects»
(model of set theory), means describing everything, this
«everything» that is described, is only at any time the current
universe, the one of our past ; our act of interpreting
expressions there, forms our present beyond this past. And then,
describing our previous act of description, means adding to this
previous description (this «everything» described) something else
beyond it.
The infinite time between models
As a one-model
theory T' describes a theory T with a model M,
the components (notions and structures)
of the model [T,M] of T', actually fall into 3
categories:
- The components of T and its developments as a formal
system (abstract types, structure symbols, expressions, axioms,
proofs from axioms), that aim to describe the model but remain
outside it and independent of it.
- The components of M (interpretations of types and
structure symbols)
- The interpretation (attribution of values) of all expressions
of T in M, for any values of their free
variables.
This last part of [T,M] is a mathematical construction
determined by the combination of both systems T and M
but it is not directly contained in them : it is built after
them.
So, the model [T,M] of T', encompassing the
present theory T with the interpretation of all its formulas
in the present universe M of past objects, is the next universe
of the past, which will come as the infinity of all current
interpretations (in M) of formulas of T will become
past.
Or can it be otherwise ? Would it be possible for a theory T
to express or simulate the notion of its own formulas and
compute their values ?
Other languages:
FR : Temps en
théorie des modèles